lecture 5: utility maximization problemsd.umn.edu/~watanabe/econ460su10/doc/umpho.pdf · lecture 5:...
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Intro Solving UMP Extreme Cases Σ
Econ 4601 Urban & Regional Economics
Lecture 5: Utility Maximization Problems
Instructor: Hiroki Watanabe
Summer 2010
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Intro Solving UMP Extreme Cases Σ
1 Introduction
2 Solving UMPBudget Line Meets Indifference CurvesTangencyFind the Exact Solutions
3 Extreme CasesPerfect SubstitutesPerfect Complements
4 Summary
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Intro Solving UMP Extreme Cases Σ
So far we have discussed:$(Lecture 2)� (Lecture 3), (Lecture 4)
Now combine all the above to predict Greg’sconsumption behavior given his budget constraintand preferences.Situational background:
Greg wants to have as many cheesecakes and teaas possible.Greg’s budget constraint does not allow him tochoose (xC,xT) =(infinite, infinite).Greg has to make choices.
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Intro Solving UMP Extreme Cases Σ
We assume that Greg picks the most preferredcombination among what he can afford.The framework to analyze Greg’s choice behavior iscalled utility maximization problem (UMP for short).
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Intro Solving UMP Extreme Cases Σ
Greg’s decision making process is summarized asfollows:
Utility Maximization Problem
Greg chooses the bundle (xC,xT) that gives him thehighest utility level , among the affordable bundles. Inother words, he
maxu(xC,xT) subject to pCxC + pTxT = m.
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Intro Solving UMP Extreme Cases Σ
1 Introduction
2 Solving UMPBudget Line Meets Indifference CurvesTangencyFind the Exact Solutions
3 Extreme CasesPerfect SubstitutesPerfect Complements
4 Summary
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Intro Solving UMP Extreme Cases Σ
Budget Line Meets Indifference Curves
How exactly do we find Greg’s optimal bundlex∗ = (x∗
C,x∗
T)?
Consider the case whenu(xC,xT) = xCxTm = 60,(pC,pT) = (4,3).
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Intro Solving UMP Extreme Cases Σ
Budget Line Meets Indifference Curves
0 5 10 15 200
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Cheesecakes (xC)
Tea
(x T
)
1
Figure:
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Intro Solving UMP Extreme Cases Σ
Budget Line Meets Indifference Curves
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Cheesecakes (xC)
Tea
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)
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Figure:
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Intro Solving UMP Extreme Cases Σ
Budget Line Meets Indifference Curves25
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Cheesecakes (xC)
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Figure:
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Intro Solving UMP Extreme Cases Σ
Budget Line Meets Indifference Curves
Figure:
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Intro Solving UMP Extreme Cases Σ
Tangency
Tangency Condition
At the optimal bundle, the indifference curve is tangentto the budget constraint (for standard preferences), i.e.,they both have the same slope at x∗.
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Intro Solving UMP Extreme Cases Σ
Tangency
What does the tangency condition imply?The slope of IC denotes MRS (MWTP): the cups oftea Greg is willing to give up for one slice ofcheesecake.The slope of budgte line denotes relative price (op.cost): the cups of tea Greg has to give up for oneslice of cheesecake.Greg’s idea of the tea’s worth coincides withmarket’s idea of the tea’s worth when he choosesthe right amount.
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Intro Solving UMP Extreme Cases Σ
Tangency
What if the tangency condition is not met?Suppose MRS is larger than the relative price (IC issteeper than the budget line).
Greg is willing to give up 30 cups of tea for a slice ofcheesecake,while he has to sell only 4
3 cups of tea to buy a sliceof cheesecake.It’s wise to increase cheesecake consumption andreduce cups of tea.The current bundle is not optimal.
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Intro Solving UMP Extreme Cases Σ
Tangency
Suppose MRS is smaller than the relative price (IC issteeper than the budget line).
Greg is willing to give up .1 cups of tea for a slice ofcheesecake,while he has to sell 4
3 cups of tea to buy a slice ofcheesecake.It’s wise to reduce cheesecake consumption andincrease cups of tea.The current bundle is not optimal.
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Intro Solving UMP Extreme Cases Σ
Find the Exact Solutions
Where exactly is the bundget line tangent to theindifferent curve?We need a little bit of calculus to find Greg’soptimal bundle.
UMPmaxu(xC,xT) = xCxT subject to 4xC + 3xT = 60.
MRS at (xC,xT) is given by −xTxC
.
The relative price is −43 .
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Intro Solving UMP Extreme Cases Σ
Find the Exact Solutions
Tangency condition:
−xTxC
=−4
3⇒ xT =
4
3xC. (1)
There are lots of bundles (xC,xT) satisfying (1).The bundle also has to be affordable:
4xC + 3xT = 60⇒ 4xC + 34
3xC = 60.
xT = 7.5 and xC = 10.Conclude x∗ = (10,7.5).
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Intro Solving UMP Extreme Cases Σ
Find the Exact Solutions
ExerciseGreg spends his income ($5) on coins (xC) and tea (xT).Price is given by (pC,pT) = (2,1). His preferences arerepresented by u(xC,xT) = xC +
pxT.
1 Write down the budget line.2 Write Greg’s utility maximization problem.3 What is the tangency condition in this example?
(MRS at (xC,xT) is −2pxT).
4 Which bundle will Greg buy?
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Intro Solving UMP Extreme Cases Σ
Find the Exact Solutions
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Coins (xC)
Tea
(x T
)
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Figure:
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Intro Solving UMP Extreme Cases Σ
1 Introduction
2 Solving UMPBudget Line Meets Indifference CurvesTangencyFind the Exact Solutions
3 Extreme CasesPerfect SubstitutesPerfect Complements
4 Summary
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Intro Solving UMP Extreme Cases Σ
There are some exceptions where tangencycondition does not apply.
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Intro Solving UMP Extreme Cases Σ
Perfect Substitutes
Consider six-packs and bottles of Corona (x6,x1).Greg’s MRS is 6 everywhere.Consider three cases:
1 Market rate of exchange is higher than 6.2 Market rate of exchange is lower than 6.3 Market rate of exchange is exactly 6.
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Intro Solving UMP Extreme Cases Σ
Perfect Substitutes
1 Market rate of exchange is higher than 6.Let’s say the market rate of exchange is 24 (i.e., 24bottles of Corona is traded for 1 six-pack).Greg is willing to give up 6 bottles for 1 six-pack.Greg will spend all his income on six-packs.This type of solution is called a corner solution (e.g.,(8,0), (35,0), (0,68) etc.)
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Intro Solving UMP Extreme Cases Σ
Perfect Substitutes
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Six−Packs (x6)
Bot
tles
(x1)
Figure:
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Intro Solving UMP Extreme Cases Σ
Perfect Substitutes
2 Market rate of exchange is lower than 6.Let’s say the market rate of exchange is 1 (i.e., 1bottle of Corona is traded for 1 six-pack).Greg is willing to give up 6 bottles for 1 six-pack.Greg will spend all his income on bottles.
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Intro Solving UMP Extreme Cases Σ
Perfect Substitutes
0 1 2 3 4 5 60
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Six−Packs (x6)
Bot
tles
(x1)
Figure:
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Intro Solving UMP Extreme Cases Σ
Perfect Substitutes
3 Market rate of exchange is exactly 6.The market rate of exchange is 6 (i.e., 6 bottles ofCorona is traded for 1 six-pack).Greg is willing to give up 6 bottles for 1 six-pack.Greg can choose any bundle on his budget line.
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Intro Solving UMP Extreme Cases Σ
Perfect Substitutes
0 1 2 3 4 5 60
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Six−Packs (x6)
Bot
tles
(x1)
Figure:
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Intro Solving UMP Extreme Cases Σ
Perfect Substitutes
DiscussionIf a bottle of Corona is sold at $2, a six-pack isusually sold at less than $12.Does that mean nobody buys Corona by the bottle?
Dharma’s MRS might be less than 6.She is willing to give up 1 six-pack for 4 bottles(MRS= 4).So, there are Corona sold by the bottle, and thereare people who buy Corona by the bottle.
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Intro Solving UMP Extreme Cases Σ
Perfect Substitutes
ExerciseGreg has $12 and spend his income on Coke & Pepsi(xC,xP). Suppose Coke is sold at $4 while Pepsi is soldat $2.
1 Find Greg’s optimal bundle.2 Confirm your answer using indifference curves and
the budget line.
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Intro Solving UMP Extreme Cases Σ
Perfect Substitutes
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Coke (xC)
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si (
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Figure:
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Intro Solving UMP Extreme Cases Σ
Perfect Complements
For perfect complements, MRS is not defined.That does not mean the solution to UMP does notexist.Consider left gloves (x1) and right gloves (x2).
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Intro Solving UMP Extreme Cases Σ
Perfect Complements
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Left Gloves (x1)
Rig
ht G
love
s (x
2)
Figure:
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Intro Solving UMP Extreme Cases Σ
Perfect Complements
Optimal bundle x∗ for left gloves and right gloves is
1 on the budget line and2 on the 45 degree line.
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Intro Solving UMP Extreme Cases Σ
Perfect Complements
ExerciseConsider a bundle (cereal, milk)= (xC,xM).
Greg says he can’t have cereals without milk andthe only time he drink milk is when he eats hiscereals.Greg’s preferred cereal-milk ratio is 2 to 3.(pC,pM) = (6,4).m = 24.
1 Draw his budget line.2 Draw indifference curves at (2,3) and (4,6).3 Mark the bundle Greg will choose on your graph.
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Intro Solving UMP Extreme Cases Σ
Perfect Complements
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Cereals (xC)
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(x M
)
Figure:
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Intro Solving UMP Extreme Cases Σ
Perfect Complements
Greg’s preferred bundles are lined up on the rayxM = 3
2xC.The optimal bundle is
1 on the ray xM = 32xC and
2 on the budget line xM = −32 xC + 6.
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Intro Solving UMP Extreme Cases Σ
1 Introduction
2 Solving UMPBudget Line Meets Indifference CurvesTangencyFind the Exact Solutions
3 Extreme CasesPerfect SubstitutesPerfect Complements
4 Summary
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Intro Solving UMP Extreme Cases Σ
How to set up the utility maximization problem.Tangency condition for standard cases.Optimal bundles for extreme preferences.
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